We consider a scheduling environment with m (m ≥ 1) identical machines in parallel and two agents. Agent A is responsible for n1 jobs and has a given objective function with regard to these jobs; agent B is responsible for n2 jobs and has an objective function that may be either the same or different from the one of agent A. The problem is to find a schedule for the n1 + n2 jobs that minimizes the objective of agent A (with regard to his n1 jobs) while keeping the objective of agent B (with regard to his n2 jobs) below or at a fixed level Q. The special case with a single machine has recently been considered in the literature, and a variety of results have been obtained for two-agent models with objectives such as fmax, ∑ wjCj, and ∑ Uj. In this paper, we generalize these results and solve one of the problems that had remained open. Furthermore, we enlarge the framework for the two-agent scheduling problem by including the total tardiness objective, allowing for preemptions, and considering jobs with different release dates; we consider also identical machines in parallel. We furthermore establish the relationships between two-agent scheduling problems and other areas within the scheduling field, namely rescheduling and scheduling subject to availability constraints.

Background Market-applicable concurrent electrocardiogram (ECG) diagnosis for multiple heart abnormalities that covers a wide range of arrhythmias, with better-than-human accuracy, has not yet been developed. We therefore aimed to engineer a deep learning approach for the automated multilabel diagnosis of heart rhythm or conduction abnormalities by real-time ECG analysis. MethodsWe used a dataset of ECGs (standard 10 s, 12-channel format) from adult patients (aged ≥18 years), with 21 distinct rhythm classes, including most types of heart rhythm or conduction abnormalities, for the diagnosis of arrhythmias at multilabel level. The ECGs were collected from three campuses of Tongji Hospital (Huazhong University of Science and Technology, Wuhan, China) and annotated by cardiologists. We used these datasets to develop a convolutional neural network approach to generate diagnoses of arrythmias. We collected a test dataset of ECGs from a new group of patients not included in the training dataset. The test dataset was annotated by consensus of a committee of board-certified, actively practicing cardiologists. To evaluate the performance of the model we assessed the F1 score and the area under the curve (AUC) of the receiver operating characteristic (ROC) curve, as well as quantifying sensitivity and specificity. To validate our results, findings for the test dataset were compared with diagnoses made by 53 ECG physicians working in cardiology departments who had a wide range of experience in ECG interpretation (range 0 to >12 years). An external public validation dataset of 962 ECGs from other hospitals was used to study generalisability of the diagnostic model.

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